Having a little trouble deriving a result in a book.(adsbygoogle = window.adsbygoogle || []).push({});

If I have an operator of the form [tex]e^{\alpha A \otimes I_n}[/tex]

Where alpha is a complex constant, A a square hermitian matrix and I the identity matrix.

Now if I want to operator that on a tensor product, say for instance [tex]c_{n,1} |1 \rangle \otimes |n \rangle[/tex] then how would I do that?

I firstly used the identity [tex]e^{\alpha A \otimes I_n} = e^{\alpha A \otimes I_n} [/tex] to obtain:

[tex]e^{\alpha A \otimes I_n} {} | c_{n,1} |1 \rangle \otimes |n \rangle \rangle = e^{\alpha A} \otimes I_n {} |c_{n,1} {}|1\rangle \otimes |n \rangle \rangle = c_{n,1} e^{\alpha A}{}|1\rangle \otimes |n \rangle [/tex]

but my book gets [tex]c_{n,1} e^{\alpha}|1 \rangle \otimes |n \rangle [/tex] with no further explanation. By the way the form of the matrix [tex] A = \begin{pmatrix} 1 & 0 & 0 \\0 & 0 & 0 \\0 & 0 & -1 \end{pmatrix} [/tex] if it helps to know it.

Also I am aware you can represent [tex]e^{\alpha A}[/tex] in the form of an infinite series but I don't see how that helps here. In fact I tried it and I didn't know where I should cut the series off at, and it gave me coefficients of alpha rather than e^alpha. Oh and the n's are being summed over, not sure if that makes any difference.

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# Unitary Operator on a Tensor Product

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